5.3.3.2.1.

Latin square and related designs

Latin square designs, and the related Graeco-Latin square and
Hyper-Graeco-Latin square designs, are a special type of comparative
design.

There is a single factor of primary interest, typically called the
treatment factor, and several nuisance factors. For Latin square
designs there are 2 nuisance factors, for Graeco-Latin square designs
there are 3 nuisance factors, and for Hyper-Graeco-Latin square designs
there are 4 nuisance factors.

Nuisance factors used as blocking variables

The nuisance factors are used as blocking variables.

For Latin square designs, the 2 nuisance factors are divided
into a tabular grid with the property that each row and each
column receive each treatment exactly once.

As with the Latin square design, a Graeco-Latin square design is
a kxk tabular grid in which k is the number
of levels of the treatment factor. However, it uses 3 blocking
variables instead of the 2 used by the standard Latin square
design.

A Hyper-Graeco-Latin square design is also a kxk
tabular grid with k denoting the number of levels of the
treatment factor. However, it uses 4 blocking variables instead
of the 2 used by the standard Latin square design.

Advantages and disadvantages of Latin square designs

The advantages of Latin square designs are:

They handle the case when we have several nuisance factors and
we either cannot combine them into a single factor or we
wish to keep them separate.

They allow experiments with a relatively small number
of runs.

The disadvantages are:

The number of levels of each blocking variable must
equal the number of levels of the treatment factor.

The Latin square model assumes that there are no interactions
between the blocking variables or between the treatment variable
and the blocking variable.

Models for Graeco-Latin and Hyper-Graeco-Latin squares are the obvious
extensions of the Latin square model, with additional blocking variables
added.

Estimates for Latin Square Designs

Estimates

Estimate for
\( \mu \):

\( \bar{Y} \) = the average of all the data

Estimate for
Ri:

\( \bar{Y}_{i} - \bar{Y} \)

\( \bar{Y}_{i} \) = average of all Y for which
X1 = i

Estimate for
Cj:

\( \bar{Y}_{j} - \bar{Y} \)

\( \bar{Y}_{j} \) = average of all Y for which
X2 = j

Estimate for
Tk:

\( \bar{Y}_{k} - \bar{Y} \)

\( \bar{Y}_{k} \) = average of all Y for which
X3 = k

Randomize as much as design allows

Designs for Latin squares with 3-, 4-, and 5-level factors are given
next. These designs show what the treatment combinations should be for
each run. When using any of these designs, be sure to randomize
the treatment units and trial order, as much as the design
allows.

For example, one recommendation is that a Latin square design
be randomly selected from those available, then randomize the run
order.